07/03/2014, 02:17 PM
Hey everybody! Well I've boiled down my requirements for solving tetration, pentation, semi operators, and a whole list of recursive relationships using fractional calculus into a single theorem. I am pretty certain this theorem will be true.
Well I'll start by saying, if \( f(x) = \sum_{n=0}^\infty a_n x^n/n! \) where \( F(a_n) = a_{n+1} \) then under certain conditions \( F(\frac{d^{z}}{dx^z}|_{x=0}f(x)) = \frac{d^{z+1}}{dx^{z+1}}|_{x=0}f(x) \)
Now of course, the problem is that when \( a_n \) is something like tetration, or pentation, or whatever, this doesn't converge and we are stuck in the mud.
So I've boiled a way to fix this. Now I don't have this theorem yet, but if its solved, all that's required is a bunch of lemmas I know how to prove and we will have tetration, pentation, hexation, semi operators, and some more.
So without further ado, here is the theorem we need.
Assume \( a_n \) is a sequence of complex numbers such that \( f(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!} \) is entire. Then, there always exists \( b_n \) such that, \( g(x) = \sum_{n=0}^\infty b_n \frac{x^n}{n!} \) is entire and Weyl differintegrable on all of \( \mathbb{C} \) and
\( h(x) = \sum_{n=0}^\infty a_n b_n \frac{x^n}{n!} \)
is such that \( \frac{d^z}{dx^z}{|}_{x=0} h(x) \) exists for all z.
If this theorem is shown, then... define \( G(z) = \frac{\frac{d^z}{dx^z}|_{x=0} h(x)}{\frac{d^z}{dx^z}|_{x=0} g(x)} \)
and
\( F(G(z)) = G(z+1) \)
and we are done.
Any one have any advice on how I can show this theorem? this is quite a stump.
Well I'll start by saying, if \( f(x) = \sum_{n=0}^\infty a_n x^n/n! \) where \( F(a_n) = a_{n+1} \) then under certain conditions \( F(\frac{d^{z}}{dx^z}|_{x=0}f(x)) = \frac{d^{z+1}}{dx^{z+1}}|_{x=0}f(x) \)
Now of course, the problem is that when \( a_n \) is something like tetration, or pentation, or whatever, this doesn't converge and we are stuck in the mud.
So I've boiled a way to fix this. Now I don't have this theorem yet, but if its solved, all that's required is a bunch of lemmas I know how to prove and we will have tetration, pentation, hexation, semi operators, and some more.
So without further ado, here is the theorem we need.
Assume \( a_n \) is a sequence of complex numbers such that \( f(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!} \) is entire. Then, there always exists \( b_n \) such that, \( g(x) = \sum_{n=0}^\infty b_n \frac{x^n}{n!} \) is entire and Weyl differintegrable on all of \( \mathbb{C} \) and
\( h(x) = \sum_{n=0}^\infty a_n b_n \frac{x^n}{n!} \)
is such that \( \frac{d^z}{dx^z}{|}_{x=0} h(x) \) exists for all z.
If this theorem is shown, then... define \( G(z) = \frac{\frac{d^z}{dx^z}|_{x=0} h(x)}{\frac{d^z}{dx^z}|_{x=0} g(x)} \)
and
\( F(G(z)) = G(z+1) \)
and we are done.Any one have any advice on how I can show this theorem? this is quite a stump.